Common Number Patterns

Common Number Patterns

There are many different types of number pattern. They are also sometimes called sequences or progressions.

Arithmetic Sequences

Arithmetic sequences have a common difference between consecutive terms:

For example

1 4 7 10 13 16…..

these dots imply that

the series continues

Or 5 12 19 26 33 40 ……

Or 20 18 16 14 12 10 ……

Geometric Sequences

Geometric sequences have a common multiplier between consecutive terms:

For example

1 3 9 27 81 243…

Or 4 20 100 500 2500 12500 …

Or 8 4 2 1 ½ ¼ …

Fibonacci Sequences

Named after an Italian mathematician, Fibonacci sequences have each term as the sum of the previous two terms:

For example

1 1 2 3 5 8 13 …

Or 1 3 4 7 11 18 29 …

Or 2 5 7 12 19 31 50 …

Other Sequences

And there are sequences based on formulae:

1 4 9 16 25 36 49… is called the square numbers

1 8 27 64 125 216… is called the cubed numbers

2 3 5 7 11 13 17… is the prime numbers

1 4 9 12 17 20 25… is an alternating sequence (add 3, add 5, add 3, add 5, etc.)

Examples

Find the next 3 terms of the following sequences:

a. 30 35 40 45 50 …

b. 102 89 76 63 50 …

c. 1458 486 162 54 18 …

d. 4 5 9 14 23 …

e. 4 6 5 7 6 8 …

f. 2 5 10 17 26 …

a. Arithmetic sequence with a common difference of 5

55 60 65

b. Arithmetic sequence with a common difference of -13

37 24 11

c. Geometric sequence with a common multiplier of ⅓

6 2 ⅔

d. Fibonacci sequence

37 60 97

e. Alternating sequence (add 2, subtract 1, add 2, etc.)

7 9 8

f. Square numbers plus 1

37 50 65

You might notice that (f) above has differences increasing by 2, and you could use this to find the next terms, as it is a property of the square numbers.

These are the sequences that you will be expected to recognise, but there are many other (sometimes strange) sequences.